I don't think they threw James' ethnicity in our faces on Supergirl other than he was black, especially early on. I can think of more times it came up with Uhura than for James. Later as he became Guardian and when he moved on back to being a reporter there may have been subtext but I wouldn't say it was thrown in our faces.
No, it was not thrown in audiences faces, thanks to the showrunners pulling time-honored BS of using black characters as their tokens, having no interest in or clue about having a black male character's view and approach to the world be informed by who he is, which is not some act or personal choice one can don or remove like a costume. Berlanti, et al., were interested in checking off their not-so-progressive box by having Brooks cast as Olsen, without it meaning anything as a black male character. Overall, he was marginalized until he was unceremoniously dropped from this series.